982.20 Starting With Parts: The Nonradial Line: Since humanity
started with
parallel lines, planes, and cubes, it also adopted the
edge line of the square and cube as the
prime unit of mensuration. This inaugurated geomathematical
exploration and analysis
with a part of the whole, in contradistinction to synergetics'
inauguration of exploration
and analysis with total Universe, within which it discovers
whole conceptual systems,
within which it identifies subentities always dealing
with experimentally discovered and
experimentally verifiable information. Though life started
with whole Universe, humans
happened to pick one part__the line, which was so short
a section of Earth arc (and the
Earth's diameter so relatively great) that they assumed
the Earth-scratched-surface line to
be straight. The particular line of geometrical reference
humans picked happened not to be
the line of most economical interattractive integrity.
It was neither the radial line of
radiation nor the radial line of gravity of spherical
Earth. From this nonradial line of
nature's event field, humans developed their formulas
for calculating areas and volumes of
the circle and the sphere only in relation to the cube-edge
lines, developing empirically the
"transcendentally irrational," ergo incommensurable,
number pi (), 3.14159 . . . ad
infinitum, which provided practically tolerable approximations
of the dimensions of circles
and spheres.

982.21
Synergetics has discovered that the vectorially most
economical control line
of nature is in the diagonal of the cube's face and
not in its edge; that this diagonal
connects two spheres of the isotropic-vector-matrix
field; and that those spherical centers
are congruent with the two only-diagonally-interconnected
corners of the cube.
Recognizing that those cube-diagonal-connected spheres
are members of the closest
packed, allspace-coordinating, unit radius spheres field,
whose radii = 1 (unity), we see
that the isotropic-vector-matrix's field-occurring-cube's
diagonal edge has the value. of 2,
being the line interconnecting the centers of the two
spheres, with each half of the line
being the radius of one sphere, and each of the whole
radii perpendicular to the same
points of intersphere tangency.

982.30 Diagonal of Cube as Control Length: We have learned
elsewhere that the
sum of the second powers of the two edges of a right
triangle equals the second power of
the right triangle's hypotenuse; and since the hypotenuse
of the two similar equiedged
right triangles formed on the square face of the cube
by the sphere-center-connecting
diagonal has a value of two, its second power is four;
therefore, half of that four is the
second power of each of the equi-edges of the right
triangle of the cube's diagonaled face:
half of four is two.

982.31
The square root of 2 = 1.414214, ergo, the length
of each of the cube's
edges is 1.414214. The sqrt(2)happens to be one of those extraordinary
relationships of
Universe discovered by mathematics. The relationship
is: the number one is to the second
root of two as the second root of two is to two: 1:sqrt(2)
= sqrt(2):2, which, solved, reads
out as 1 : 1.414214 = 1.414214 : 2.

982.32
The cube formed by a uniform width, breadth, and height
of sqrt(2) is sqrt(23),
which = 2.828428. Therefore, the cube occurring in nature
with the isotropic vector
matrix, when conventionally calculated, has a volume
of 2.828428.

982.33
This is exploratorily noteworthy because this cube,
when calculated in terms
of man's conventional mensuration techniques, would
have had a volume of one, being the
first cube to appear in the omni-geometry-coordinate
isotropic vector matrix; its edge
length would have been identified as the prime dimensional
input with an obvious length
value of one__ergo, its volume would
be one: 1 × 1 ×
1 = 1. Conventionally calculated,
this cube with a volume of one, and an edge length of
one, would have had a face diagonal
length of sqrt(2), which equals 1.414214. Obviously, the use
of the diagonal of the cube's
face as the control length results in a much higher
volume than when conventionally
evaluated.

982.40 Tetrahedron and Synergetics Constant: And now comes
the big surprise,
for we find that the cube as coordinately reoccurring
in the isotropic vector matrix__as
most economically structured by nature__has a volume
of three in synergetics' vector-
edged, structural-system-evaluated geometry, wherein
the basic structural system of
Universe, the tetrahedron, has a volume of one.

982.42
To have its cubical conformation structurally (triangulated)
guaranteed (see
Secs.
615
and
740),
the regular equiangled tetrahedron
must be inserted into the cube,
with the tetrahedron's six edges congruent with each
of the six vacant but
omnitriangulatable diagonals of the cube's six square
faces.

982.43
As we learn elsewhere (Secs.
415.22
and
990), the
tetrahedron is not only
the basic structural system of Universe, ergo, of synergetic
geometry, but it is also the
quantum of nuclear physics and is, ipso facto, exclusively
identifiable as the unit of
volume; ergo, tetrahedron volume equals one. We also
learned in the sections referred to
above that the volume of the octahedron is exactly four
when the volume of the
tetrahedron of the unit-vector edges of the isotropic-vector-matrix
edge is one, and that
four Eighth-Octahedra are asymmetrical tetrahedra with
an equiangular triangular base,
three apex angles of 90 degrees, and six lower-comer
angles of 45 degrees each; each of
the 1/8th octahedron's asymmetric tetrahedra has a volumetric
value of one-half unity (the
regular tetrahedron). When four of the Eighth-Octahedrons
are equiangle-face added to
the equiangled, equiedged faces of the tetrahedra, they
produce the minimum cube, which,
having the tetrahedron at its heart with a volume of
one, has in addition four one-half unity
volumed Eighth-Octahedra, which add two volumetric units
on its corners. Therefore, 2 +
1 = 3 = the volume of the cube. The cube is volume three
where the tetrahedron's volume
is one, and the octahedron's volume is four, and the
cube's diagonally structured faces
have a diagonal length of one basic system vector of
the isotropic vector matrix. (See
Illus.
463.01.)

982.44
Therefore the edge of the cube = sqrt(1/2).

982.45
Humanity's conventional mensuration cube with a volume
of one turns out
in energetic reality to have a conventionally calculated
volume of 2.828428, but this same
cube in the relative-energy volume hierarchy of synergetics
has a volume of 3.

982.47
Next we discover, as the charts at Secs.
963.10
and
223.64 show, that of the
inventory of well-known symmetrical polyhedra of geometry,
all but the cube have
irrational values as calculated in the XYZ rectilinear-coordinate
system__"cubism" is a
convenient term__in which the cube's edge and volume
are both given the prime
mensuration initiating value of one. When, however,
we multiply all these irrational values
of the Platonic polyhedra by the synergetic conversion
constant, 1.06066, all these values
become unitarily or combinedly rational, and their low
first-four-prime-number-
accommodation values correspond exactly with those of
the synergetic hierarchy of
geometric polyhedra, based on the tetrahedron as constituting
volumetric unity.

982.48
All but the icosahedron and its "wife," the pentagonal
dodecahedron, prove
to be volumetrically rational. However, as the tables
show, the icosahedron and the
vector-edged cube are combiningly rational and together
have the rational value of three
to the third power, i.e., 27. We speak of the pentagonal
dodecahedron as the
icosahedron's wife because it simply outlines the surface-area
domains of the 12 vertexes
of the icosahedron by joining together the centers of
area of the icosahedron's 20 faces.
When the pentagonal dodecahedron is vectorially constructed
with flexible tendon joints
connecting its 30 edge struts, it collapses, for, having
no triangles, it has no structural
integrity. This is the same behavior as that of a cube
constructed in the same flexible-
tendon-vertex manner. Neither the cube nor the pentagonal
dodecahedron is scientifically
classifiable as a structure or as a structural system
(see Sec.
604).

982.50 Initial Four-Dimensional Modelability: The modelability
of the XYZ
coordinate system is limited to rectilinear-frame-of-reference
definition of all special-case
experience patternings, and it is dimensionally sized
by arbitrary, e.g., c.gt.s.-system,
subdivisioning increments. The initial increments are
taken locally along infinitely
extensible lines always parallel to the three sets of
rectilinearly interrelated edges of the
cube. Any one of the cube's edges may become the one-dimensional
module starting
reference for initiating the mensuration of experience
in the conventional, elementary,
energetical7 school curriculum.

(Footnote 7: Energetical is in contradistinction to synergetical.
Energetics employs isolation of special cases of our total experience, the
better to discern unique behaviors of parts undiscernible and
unmeasurable in total experience.)

982.51
The XYZ cube has no initially moduled, vertex-defined
nucleus; nor has it
any inherent, common, most-economically-distanced, uniform,
in-out-and-
circumferentially-around, corner-cutting operational
interlinkage, uniformly moduled
coordinatability. Nor has it any initial, ergo inherent,
time-weight-energy-(as mass charge
or EMF) expressibility. Nor has it any omni-intertransformability
other than that of vari-
sized cubism. The XYZ exploratory coordination inherently
commences differentially, i.e.,
with partial system consideration. Consider the three-dimensional,
weightless, timeless,
temperatureless volume often manifest in irrational
fraction increments, the general reality
impoverishments of which required the marriage of the
XYZ system with the c.gt.s. system
in what resembles more of an added partnership than
an integration of the two.

982.53
In the synergetics' four-, five-, and six-dimensionally
coordinate system's
operational field the linear increment modulatability
and modelability is the isotropic
vector matrix's vector, with which the edges of the
co-occurring tetrahedra and octahedra
are omnicongruent; while only the face diagonals__and
not the edges__of the inherently
co-occurring cubes are congruent with the matrix vectors.
Synergetics' exploratory
coordination inherently commences integrally, i.e.,
with whole-systems consideration.
Consider the one-dimensional linear values derived from
the initially stated whole system,
six-dimensional, omnirational unity; any linear value
therefrom derived can be holistically
attuned by unlimited frequency and one-to-one, coordinated,
wavelength modulatability.
To convert the XYZ system's cubical values to the synergetics'
values, the mathematical
constants are linearly derived from the mathematical
ratios existing between the
tetrahedron's edges and the cube's corner-to-opposite-corner
distance relationships; while
the planar area relationships are derived from the mathematical
ratios existing between
cubical-edged square areas and cubical-face-diagonaled-edged
triangular areas; and the
volumetric value mathematical relationships are derived
from ratios existing between (a)
the cube-edge-referenced third power of the-often odd-fractioned-edge
measurements
(metric or inches) of cubically shaped volumes and (b)
the cube-face-diagonal-vector-
referenced third power of exclusively whole number vector,
frequency modulated,
tetrahedrally shaped volumes. (See Sec.
463
and
464
for exposition of the diagonal of the
cube as a wave-propagation model.)

982.54
The mathematical constants for conversion of the linear,
areal, and
volumetric values of the XYZ system to those of the
synergetics system derive from the
synergetics constant (1.060660). (See Sec.
963.10
and
Chart
963.12.) The conversion
constants are as follows:

First Dimension: The first dimensional cube-edge-to-cube-face-diagonal
vector
conversion constant from XYZ to synergetics is as 1:1.060660.

982.55
To establish a numerical value for the sphere, we
must employ the
synergetics constant for cubical third-power volumetric
value conversion of the vector
equilibrium with the sphere of radius 1. Taking the
vector equilibrium at the initial phase
(zero frequency, which is unity-two diameter: ergo unity-one
radius) with the sphere of
radius l; i.e., with the external vertexes of the vector
equilibrium congruent with the
surface of the sphere = 4/3 pi ()
multiplied by the
third power of the radius. Radius = 1.
13 = 1. l × 1.333 × 3.14159 = 4.188. 4.188 times synergetics
third-power constant 1.192
= 5 = volume of the sphere. The volume of the radius
1 vector equilibrium = 2.5. VE
sphere = 2 VE.

982.56
We can assume that when the sphere radius is 1 (the
same as the nuclear
vector equilibrium) the Basic Disequilibrium 120 LCD
tetrahedral components of mild off-
sizing are also truly of the same volumetric quanta
value as the A and B Quanta Modules;
they would be shortened in overall greatest length while
being fractionally fattened at their
smallest-triangular-face end, i.e., at the outer spherical
surface end of the 120 LCD
asymmetric tetrahedra. This uniform volume can be maintained
(as we have seen in Sec.
961.40).

982.57
Because of the fundamental 120-module identity of
the nuclear sphere of
radius 1 (F = 0), we may now identify the spherical
icosahedron of radius 1 as five; or as
40 when frequency is 2F2. Since 40 is also the volume
of the F2 vector-equilibrium-
vertexes-congruent sphere, the unaberrated vector equilibrium
F2 = 20 (i.e., 8 × 2 1/2
nuclear-sphere's inscribed vector equilibrium). We may
thus assume that the spherical
icosahedron also subsides by loss of half its volume
to a size at which its volume is also
20, as has been manifested by its prime number five,
indistinguishable from the vector
equilibrium in all of its topological hierarchies characteristics.

982.58
Neither the planar-faceted exterior edges of the icosahedron
nor its radius
remain the same as that of the vector equilibrium, which,
in transforming from the vector
equilibrium conformation to the icosahedral state__as
witnessed in the jitterbugging (see
Sec.
465)
__did so by transforming its outer edge lengths
as well as its radius. This
phenomenon could be analagous the disappearance of the
nuclear sphere, which is
apparently permitted by the export of its volume equally
to the 12 surrounding spheres
whose increased diameters would occasion the increased
sizing of the icosahedron to
maintain the volume 20-ness of the vector equilibrium.
This supports the working
assumption that the 120 LCD asymmetric tetrahedral volumes
are quantitatively equal to
the A or B Quanta Modules, being only a mild variation
of shape. This effect is confirmed
by the discovery that 15 of the 120 LCD Spherical Triangles
equally and interiorly
subdivide each of the eight spherical octahedron's triangular
surfaces, which spherical
octahedron is described by the three-great-circle set
of the 25 great circles of the spherical
vector equilibrium.

982.59
We may also assume that the pentagonal-faced dodecahedron,
which is
developed on exactly the same spherical icosahedron,
is also another transformation of the
same module quantation as that of the icosahedron's
and the vector equilibrium's prime
number five topological identity.

982.60
Without any further developmental use of pi () we
may now state in
relation to the isotropic vector matrix synergetic system,
that:

The volume of the sphere is a priori always quantitatively:

__

5F3 as volumetrically referenced to the regular tetrahedron
(as volume = 1); or

982.61
There is realized herewith a succession of concentric,
12-around-one,
closest-packed spheres, each of a tetra volume of five;
i.e., of 120 A and B Quanta
Modules omniembracing our hierarchy of nuclear event
patternings. See Illus.
982.61
in
the color section, which depicts the synergetics isometric
of the isotropic vector matrix
and its omnirational, low-order whole number, equilibrious
state of the micro-macro
cosmic limits of nuclearly unique, symmetrical morphological
relativity in their
interquantation, intertransformative, intertransactive,
expansive-contractive, axially-
rotative, operational field. This may come to be identified
as the unified field, which, as an
operationally transformable complex, is conceptualizable
only in its equilibrious state.

982.61A Cosmic Hierarchy of Omnidirectionally-phased Nuclear-centered,
Convergently-divergently Intertransformable Systems:
There is realized herewith a
succession of concentric, 12-around-one, closest-packed
spheres omniembracing our
hierarchy of nuclear event patternings. The synergetics
poster in color plate 9 depicts the
synergetics isometric of the isotropic vector matrix
and its omnirational, low-order-whole-
number, equilibrious state of the macro-micro cosmic
limits of nuclearly unique,
symmetrical morphological relativity in their interquantation,
intertransformative,
intertransactive, expansive-contractive, axially rotative,
operational field. This may come
to be identified as the unified field, which, as an
operationally transformable complex, is
conceptualized only in its equilibrious state.

982.63 Sphere and Vector Equilibrium: Sphere = vector equilibrium
in combined
four-dimensional orbit and axial spin. Its 12 vertexes
describing six great circles and six
axes. All 25 great circles circling while spinning on
one axis produce a spin-profiling of a
superficially perfect sphere.

982.64
The vector equilibrium also has 25 great circles (see
Sec.
450.10), of which
12 circles have 12 axes of spin, four great circles
have four axes of spin, six great circles
have six axes of spin, and three great circles have
three axes of spin. (12 + 4 + 6 + 3 = 25)

982.70 Hierarchy of Concentric Symmetrical Geometries: It
being
experimentally demonstrable that the number of A and
B Quanta Modules per tetrahedron
is 24 (see Sec.
942.10); that the number of quanta modules
of all the symmetric polyhedra
congruently co-occurring within the isotropic vector
matrix is always 24 times their whole
regular-tetrahedral-volume values; that we find the
volume of the nuclear sphere to be five
(it has a volumetric equivalence of 120 A and B Quanta
Modules); that the common prime
number five topological and quanta-module value identifies
both the vector equilibrium
and icosahedron (despite
their exclusively unique morphologies__see
Sec.
905, especially
905.55;
that the icosahedron is one of the three-and-only
prime structural systems of
Universe (see Secs.
610.20
and
1011.30) while the vector
equilibrium is
unstable__because equilibrious__and is not a structure;
that their quanta modules are of
equal value though dissimilar in shape; and that though
the vector equilibrium may be
allspace-fillingly associated with tetrahedra and octahedra,
the icosahedron can never be
allspace-fillingly compounded either with itself nor
with any other polyhedron: these
considerations all suggest the relationship of the neutron
and the proton for, as with the
latter, the icosahedron and vector equilibrium are interexchangingly
transformable through
their common spherical-state omnicongruence, quantitatively
as well as morphologically.

982.71
The significance of this unified field as defining
and embracing the minimum-
maximum limits of the inherent nuclear domain limits
is demonstrated by the nucleus-
concentric, symmetrical, geometrical hierarchy wherein
the rhombic dodecahedron
represents the smallest, omnisymmetrical, selfpacking,
allspace-filling, six-tetra-volume,
uniquely exclusive, cosmic domain of each and every
closest-packed, unit-radius sphere.
Any of the closest-packed, unit-radius spheres, when
surrounded in closest packing by 12
other such spheres, becomes the nuclear sphere, to become
uniquely embraced by four
successive layers of surrounding, closest-packed, unit-radius
spheres__each of which four
layers is uniquely related to that nucleus__with each
additional layer beyond four
becoming duplicatingly repetitive of the pattern of
unique surroundment of the originally
unique, first four, concentric-layered, nuclear set.
It is impressive that the unique nuclear
domain of the rhombic dodecahedron with a volume of
six contains within itself and in
nuclear concentric array:

__

the unity-one-radiused sphere of volume five;

__

the octahedron of volume four;

__

the cube of volume three;

__

the prime vector equilibrium of volume 2 l/2; and

__

the two regular (positive and negative) tetrahedra
of volume one each.

This succession of 1, 2, 3, 4, 5, 6 rational volume
relationships embraces the first four
prime numbers 1, 2, 3, and 5. (See Illus.
982.61 in
color section.) The volume-24 (tetra)
cube is the largest omnisymmetrical self-packing, allspace-filling
polyhedron that exactly
identifies the unique domain of the original 12-around-one,
nuclear-initiating, closest
packing of unit-radius spheres. The unit quantum leap
of 1__going to 2__going to
3__going to 4__going to 5__going to 6, with no step greater
than 1, suggests a unique
relationship of this set of six with the sixness of
degrees of freedom.8

(Footnote 8: For further suggestions of the relationship between the rhombic
dodecahedron and the degrees of freedom see Sec.
426537.10954.47.)

982.72
The domain limits of the hierarchy of concentric,
symmetrical geometries
also suggests the synergetic surprise of two balls having
only one interrelationship; while
three balls have three__easily predictable__relationships;
whereas the simplest, ergo
prime, structural system of Universe defined exclusively
by four balls has an unpredictable
(based on previous experience) sixness of fundamental
interrelationships represented by
the six edge vectors of the tetrahedron.

982.73
The one-quantum "leap" is also manifest when one vector
edge of the
volume 4 octahedron is rotated 90 degrees by disconnecting
two of its ends and
reconnecting them with the next set of vertexes occurring
at 90 degrees from the
previously interconnected-with vertexes, transforming
the same unit-length, 12-vector
structuring from the octahedron to the first three-triple-bonded-together
(face-to-face)
tetrahedra of the tetrahelix of the DNA-RNA formulation.
One 90-degree vector
reorientation in the complex alters the volume from
exactly 4 to exactly 3. This
relationship of one quantum disappearance coincident
to the transformation of the nuclear
symmetrical octahedron into the asymmetrical initiation
of the DNA-RNA helix is a
reminder of the disappearing-quanta behavior of the
always integrally end-cohered
jitterbugging transformational stages from the 20 tetrahedral
volumes of the vector
equilibrium to the octahedron's 4 and thence to the
tetrahedron's 1 volume. All of these
stages are rationally concentric in our unified operational
field of 12-around-one closest-
packed spheres that is only conceptual as equilibrious.
We note also that per each sphere
space between closest-packed spheres is a volume of
exactly one tetrahedron: 6 - 5 = 1.